Sequences of identities involving powers of R, T, V
R-T-V evenR-T-V odd
 R −8 + T −8 + V −8 = 650 R −6 + T −6 + V −6 = 129 R −4 + T −4 + V −4 = 26 R −2 + T −2 + V −2 = 6 R 0 + T 0 + V 0 = 3 R +2 + T +2 + V +2 = 5 R +4 + T +4 + V +4 = 13 R +6 + T +6 + V +6 = 38 R +8 + T +8 + V +8 = 117
 R −7 − T −7 + V −7 = 289 R −5 − T −5 + V −5 = 57 R −3 − T −3 + V −3 = 11 R −1 − T −1 + V −1 = 2 R +1 − T +1 + V +1 = 1 R +3 − T +3 + V +3 = 4 R +5 − T +5 + V +5 = 16 R +7 − T +7 + V +7 = 57 R +9 − T +9 + V +9 = 193
RT-TV-VR evenRT-TV-VR odd
 (RT) −8 + (TV) −8 + (VR) −8 = 117 (RT) −6 + (TV) −6 + (VR) −6 = 38 (RT) −4 + (TV) −4 + (VR) −4 = 13 (RT) −2 + (TV) −2 + (VR) −2 = 5 (RT) 0 + (TV) 0 + (VR) 0 = 3 (RT) +2 + (TV) +2 + (VR) +2 = 6 (RT) +4 + (TV) +4 + (VR) +4 = 26 (RT) +6 + (TV) +6 + (VR) +6 = 129 (RT) +8 + (TV) +8 + (VR) +8 = 650
 (RT) −9 + (TV) −9 − (VR) −9 = 193 (RT) −7 + (TV) −7 − (VR) −7 = 57 (RT) −5 + (TV) −5 − (VR) −5 = 16 (RT) −3 + (TV) −3 − (VR) −3 = 4 (RT) −1 + (TV) −1 − (VR) −1 = 1 (RT) +1 + (TV) +1 − (VR) +1 = 2 (RT) +3 + (TV) +3 − (VR) +3 = 11 (RT) +5 + (TV) +5 − (VR) +5 = 57 (RT) +7 + (TV) +7 − (VR) +7 = 289
casesequencerecursive formulaOEIS
R-T-V even
RT-TV-VR even
… 129, 26, 6, 3, 5, 13, 38 … an = 5an−1 − 6an−2 + an−3 OEIS A198636
R-T-V odd
RT-TV-VR odd
… 57, 11, 2, 1, 4, 16, 57 … Alternate terms of OEIS A096975
The reverse of the recursive formula is:

αn = 6αn−1 − 5αn−2 + αn−3

which will always provide integer outputs for integer inputs. Thus it is not surprising that the R-T-V sequences succeed in yielding septrights for negative powers.

Sequences of identities involving powers of Q, S, U
Q-S-U evenQ-S-U odd
 Q 0 + S 0 + U 0 = 3 Q 2 + S 2 + U 2 = 7 Q 4 + S 4 + U 4 = 21 Q 6 + S 6 + U 6 = 70 Q 8 + S 8 + U 8 = 245 Q 10 + S 10 + U 10 = 882 Q 12 + S 12 + U 12 = 3,234
 Q 1 + S 1 − U 1 = √7 Q 3 + S 3 − U 3 = 4 √7 Q 5 + S 5 − U 5 = 14 √7 Q 7 + S 7 − U 7 = 49 √7 Q 9 + S 9 − U 9 = 175 √7 Q11 + S11 − U11 = 637 √7 Q13 + S13 − U13 = 2,352 √7
QS-SU-UQ evenQS-SU-UQ odd
 (QS) 0 + (SU) 0 + (UQ) 0 = 3 (QS) 2 + (SU) 2 + (UQ) 2 = 14 (QS) 4 + (SU) 4 + (UQ) 4 = 98 (QS) 6 + (SU) 6 + (UQ) 6 = 833 (QS) 8 + (SU) 8 + (UQ) 8 = 7,546 (QS) 10 + (SU) 10 + (UQ) 10 = 69,629 (QS) 12 + (SU) 12 + (UQ) 12 = 645,869
 (QS) 1 − (SU) 1 − (UQ) 1 = 0 (QS) 3 − (SU) 3 − (UQ) 3 = 21 (QS) 5 − (SU) 5 − (UQ) 5 = 245 (QS) 7 − (SU) 7 − (UQ) 7 = 2,401 (QS) 9 − (SU) 9 − (UQ) 9 = 22,638 (QS) 11 − (SU) 11 − (UQ) 11 = 211,288 (QS) 13 − (SU) 13 − (UQ) 13 = 1,966,419
casesequencerecursive formulaOEIS
Q-S-U even 2, 2, 3, 7, 21, 70, 245 … bn = 7bn−1 − 14bn−2 + 7bn−3 Multiply OEIS A122068 by 7
Q-S-U odd 0, 1, 4, 14, 49, 175, 637 … OEIS A215493
The reverse of the recursive formula above is:

βn = 2βn−1 − 1βn−2 + 17 βn−3

which involves division by 7, meaning that there is no guarantee that integer inputs will give rise to integer outputs. This helps explain why the Q-S-U sequences do not always give integer results for negative exponents. The same idea applies to the formula below, which when reversed will divide by 49.

QS-SU-UQ even 3, 14, 98, 833, 7546, 69629, 645869 … cn = 14cn−1 − 49cn−2 + 49cn−3 Alternate terms of OEIS A275831
QS-SU-UQ odd 0, 21, 245, 2401, 22638, 211288, 1966419

rtv3 ÷ rtv3 — alternate terms of OEIS A274220 and OEIS A215076
RT-TV-VR evenRT-TV-VR odd
 (R÷T) −8 + (T÷V) −8 + (V÷R) −8 = 72,229 (R÷T) −6 + (T÷V) −6 + (V÷R) −6 = 4,406 (R÷T) −4 + (T÷V) −4 + (V÷R) −4 = 269 (R÷T) −2 + (T÷V) −2 + (V÷R) −2 = 17 (R÷T) 0 + (T÷V) 0 + (V÷R) 0 = 3 (R÷T) +2 + (T÷V) +2 + (V÷R) +2 = 10 (R÷T) +4 + (T÷V) +4 + (V÷R) +4 = 66 (R÷T) +6 + (T÷V) +6 + (V÷R) +6 = 493 (R÷T) +8 + (T÷V) +8 + (V÷R) +8 = 3,818
 (R÷T) −7 + (T÷V) −7 − (V÷R) −7 = −17,839 (R÷T) −5 + (T÷V) −5 − (V÷R) −5 = −1,088 (R÷T) −3 + (T÷V) −3 − (V÷R) −3 = −66 (R÷T) −1 + (T÷V) −1 − (V÷R) −1 = −3 (R÷T) +1 + (T÷V) +1 − (V÷R) +1 = +4 (R÷T) +3 + (T÷V) +3 − (V÷R) +3 = +25 (R÷T) +5 + (T÷V) +5 − (V÷R) +5 = +179 (R÷T) +7 + (T÷V) +7 − (V÷R) +7 = +1,369 (R÷T) +9 + (T÷V) +9 − (V÷R) +9 = +10,672

qsu3 ÷ qsu3 — alternate terms of OEIS A033304 and OEIS A094648
QU-SQ-US evenQU-SQ-US odd
 (Q÷U) −8 + (S÷Q) −8 + (U÷S) −8 = 117 (Q÷U) −6 + (S÷Q) −6 + (U÷S) −6 = 38 (Q÷U) −4 + (S÷Q) −4 + (U÷S) −4 = 13 (Q÷U) −2 + (S÷Q) −2 + (U÷S) −2 = 5 (Q÷U) 0 + (S÷Q) 0 + (U÷S) 0 = 3 (Q÷U) +2 + (S÷Q) +2 + (U÷S) +2 = 6 (Q÷U) +4 + (S÷Q) +4 + (U÷S) +4 = 26 (Q÷U) +6 + (S÷Q) +6 + (U÷S) +6 = 129 (Q÷U) +8 + (S÷Q) +8 + (U÷S) +8 = 650
 (Q÷U) −7 + (S÷Q) −7 − (U÷S) −7 = 57 (Q÷U) −5 + (S÷Q) −5 − (U÷S) −5 = 16 (Q÷U) −3 + (S÷Q) −3 − (U÷S) −3 = 4 (Q÷U) −1 + (S÷Q) −1 − (U÷S) −1 = 1 (Q÷U) +1 + (S÷Q) +1 − (U÷S) +1 = 2 (Q÷U) +3 + (S÷Q) +3 − (U÷S) +3 = 11 (Q÷U) +5 + (S÷Q) +5 − (U÷S) +5 = 57 (Q÷U) +7 + (S÷Q) +7 − (U÷S) +7 = 289 (Q÷U) +9 + (S÷Q) +9 − (U÷S) +9 = 1460

qsu3 ÷ rtv3
QV-ST-UR evenOEIS A108716 QV-ST-UR oddOEIS A215794
 (Q÷V) 0 + (S÷T) 0 + (U÷R) 0 = 3 (Q÷V) 2 + (S÷T) 2 + (U÷R) 2 = 21 (Q÷V) 4 + (S÷T) 4 + (U÷R) 4 = 371 (Q÷V) 6 + (S÷T) 6 + (U÷R) 6 = 7,077 (Q÷V) 8 + (S÷T) 8 + (U÷R) 8 = 135,779 (Q÷V) 10 + (S÷T) 10 + (U÷R) 10 = 2,606,261 (Q÷V) 12 + (S÷T) 12 + (U÷R) 12 = 50,028,755
 (Q÷V) 1 − (S÷T) 1 − (U÷R) 1 = √7 (Q÷V) 3 − (S÷T) 3 − (U÷R) 3 = 31 √7 (Q÷V) 5 − (S÷T) 5 − (U÷R) 5 = 609 √7 (Q÷V) 7 − (S÷T) 7 − (U÷R) 7 = 11,711 √7 (Q÷V) 9 − (S÷T) 9 − (U÷R) 9 = 224,833 √7 (Q÷V) 11 − (S÷T) 11 − (U÷R) 11 = 4,315,871 √7 (Q÷V) 13 − (S÷T) 13 − (U÷R) 13 = 82,846,113 √7
QR-SV-UT even QR-SV-UT odd
 (Q÷R) 0 + (S÷V) 0 + (U÷T) 0 = 3 (Q÷R) 2 + (S÷V) 2 + (U÷T) 2 = 14 (Q÷R) 4 + (S÷V) 4 + (U÷T) 4 = 154 (Q÷R) 6 + (S÷V) 6 + (U÷T) 6 = 1,883 (Q÷R) 8 + (S÷V) 8 + (U÷T) 8 = 23,226 (Q÷R) 10 + (S÷V) 10 + (U÷T) 10 = 286,699 (Q÷R) 12 + (S÷V) 12 + (U÷T) 12 = 3,539,221
 (Q÷R) 1 + (S÷V) 1 + (U÷T) 1 = 2 √7 (Q÷R) 3 + (S÷V) 3 + (U÷T) 3 = 17 √7 (Q÷R) 5 + (S÷V) 5 + (U÷T) 5 = 203 √7 (Q÷R) 7 + (S÷V) 7 + (U÷T) 7 = 2,499 √7 (Q÷R) 9 + (S÷V) 9 + (U÷T) 9 = 30,842 √7 (Q÷R) 11 + (S÷V) 11 + (U÷T) 11 = 380,730 √7 (Q÷R) 13 + (S÷V) 13 + (U÷T) 13 = 4,700,031 √7

The recursion formula for both QR-SV-UT sequences is:

an = 14an−1 − 21an−2 + 7an−3

At the time the present report was prepared, OEIS did not index any order-3 linear recurrences with those coefficients, even those that might have other initial values.

Sequence OEIS A185963 is the closest that the present author found. It divides the coefficients by 7, resulting in the following recursion formula:

an = 2an−1 − 3an−2 + an−3